Question: Is ${495963}$ divisible by $3$ ?
Answer: A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {495963}= &&{4}\cdot100000+ \\&&{9}\cdot10000+ \\&&{5}\cdot1000+ \\&&{9}\cdot100+ \\&&{6}\cdot10+ \\&&{3}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {495963}= &&{4}(99999+1)+ \\&&{9}(9999+1)+ \\&&{5}(999+1)+ \\&&{9}(99+1)+ \\&&{6}(9+1)+ \\&&{3} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {495963}= &&\gray{4\cdot99999}+ \\&&\gray{9\cdot9999}+ \\&&\gray{5\cdot999}+ \\&&\gray{9\cdot99}+ \\&&\gray{6\cdot9}+ \\&& {4}+{9}+{5}+{9}+{6}+{3} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${495963}$ is divisible by $3$ if ${ 4}+{9}+{5}+{9}+{6}+{3}$ is divisible by $3$ Add the digits of ${495963}$ $ {4}+{9}+{5}+{9}+{6}+{3} = {36} $ If ${36}$ is divisible by $3$ , then ${495963}$ must also be divisible by $3$ Add the digits of ${36}$ $ {3}+{6} = \color{#9D38BD}{9} $ If $\color{#9D38BD}{9}$ is divisible by $3$ , then ${36}$ must also be divisible by $3$ $\color{#9D38BD}{9}$ is divisible by $3$, therefore ${495963}$ must also be divisible by $3$.